A major question in calculus of variations is to formulate an appropriate set of sufficient conditions on f and ... that the terms weak and strong local minimizers are standard in the calculus of variations and refer to ...... R. Adams. Sobolev spaces ...
P roceedings o f th e Ro ya l S ociety o f E d inbu rgh , 131A , 155{184, 20 0 1
Su± ciency theorems for local minimizers of the multiple integrals of the calculus of variations Ali Taheri¤ Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012, USA (MS received 19 October 1998; accepted 10 November 1999) Let « » Rn be a bounded domain and let f : « £ RN £ RN £ functional I(u) := f (x; u; Du) dx;
n
!
R. Consider the
«
over the class of Sobolev functions W 1;q (« ; RN ) (1 q 1 ) for which the integral on the right is well de¯ned. In this paper we establish su± cient conditions on a given function u 0 and f to ensure that u 0 provides an L r local minimizer for I where 1 r 1 . The case r = 1 is somewhat known and there is a considerable literature on the subject treating the case min(n; N ) = 1, mostly based on the ¯eld theory of the calculus of variations. The main contribution here is to present a set of su± cient conditions for the case 1 r < 1 . Our proof is based on an indirect approach and is largely motivated by an argument of Hestenes relying on the concept of `directional convergence’ .
1. Introduction Let « » Rn be a bounded domain (open connected set) and let f : « RN £n ! R. In this paper we consider functionals of the form Z I(u) = f (x; u; Du) dx;
£ RN £
(1.1)
«
over the class of admissible functions F q := fu 2 W 1;q (« ; RN ) : the integral (1.1) is well de nedg;
(1.2)
where 1 6 q 6 1. By well de ned, we mean that the integrand is a measurable function on « and that at least one of the functions f + = maxff(¢; u(¢); Du(¢)); 0g or f = minff (¢; u(¢); Du(¢)); 0g has a nite integral. It is therefore clear that · := R [ f 1; +1g. The spaces W 1;q (« ; RN ) appearing in (1.2) are the I : Fq ! R usual Sobolev spaces of vector-valued functions de ned over « and the terminology we use in this paper is in accordance with [1, 13, 36]. Throughout this paper we assume that « has a Lipschitz boundary @« with @« = @« 1 [ @« 2 [ N , where @« 1 and @« 2 are disjoint relatively open subsets of @« and Hn 1 (N ) = 0. Here, H n 1 (¢) stands for the (n 1)-dimensional Hausdor¬ measure. We denote the unit outward normal to the boundary at a point x by ¤Present address: Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK.
155
A. Taheri
156
¸ (x). Let us also mention that unless otherwise stated we shall use the summation convention on the indices. A major question in calculus of variations is to formulate an appropriate set of su¯ cient conditions on f and a given u0 2 F q to ensure that u0 provides a local minimizer for I. Of course, the notion of a local minimizer depends very much on the choice of the topology. To make this clear let us x u0 2 F q and @« 1 as described above and set Aqu0 (@«
1)
:= fu 2 F q : (u
u0 )j@«
= 0g;
1
where the boundary values are to be interpreted in the sense of traces. We now proceed by giving the following de nition. Definition 1.1. Let 1 6 r 6 1. The point u0 2 F q is an Lr (respectively, W 1;r ) local minimizer of I if and only if there exists " > 0 such that I(u0 ) 6 I(u) for all u 2 Aqu0 (@« ku
1)
satisfying
u0 kLr («
;RN )
0;
rst for all variations ’ 2 C 1 (« · ; RN ) satisfying ’j@« 1 = 0 and then by a density argument for all ’ 2 W 1;2 (« ; RN ) satisfying ’j@« 1 = 0. Condition (i), known as the Euler{Lagrange equation (or the equation of rst variation), is equivalent to u0 being a weak solution to the system @ (fPij (x; u; Du)) = fui (x; u; Du) @xj
in « ;
fPij (x; u; Du)¸ j (x) = 0
on @«
2
for i = 1; : : : ; N . We often call a solution to the above system a stationary point of I. Condition (ii) states that the quadratic form Z (fPij Pkl (x; u0 ; Du0 )’i;j ’k;l «
+ 2fPij uk (x; u0 ; Du0 )’i;j ’k + fuk ul (x; u0 ; Du0 )’k ’l ) dx is non-negative. It is well known that if this condition is slightly strengthened, that is, (ii)
there exists ® > 0 such that ¯ 2 I(u0 ; ’) > ® k’k2W 1;2 (« for all ’ 2 W
1;2
N
(« ; R ) satisfying ’j@«
1
= 0;
;RN )
Su± ciency theorems for local minimizers
157 A1u0 (@«
then (i) and (ii) would imply u0 to be a weak local minimizer in 1 ). Recall that the terms weak and strong local minimizers are standard in the calculus of variations and refer to local minimizers in W 1;1 and L1 topologies, respectively. A su¯ ciency theorem for strong local minimizers of I in the case n = N = 1 was rst properly formulated and proved by Weierstrass. His proof is based on the novel idea of constructing a ¯eld of extremals or what is known today as the eld theory of the calculus of variations (cf. [7, 8, 16]). Since then there have been numerous attempts on the one hand to extend the proof to higher dimensions, i.e. n > 1, and on the other hand to nd alternative ways to avoid the construction of such elds. In [20], Levi gave a proof for the planar case (when n = N = 1) that avoids the use of eld of extremals. Levi’s method is referred to as an expansion method since it is based on rst expanding the total variation by application of the Taylor’s formula and then showing it to be positive by the use of certain integral inequalities and properties of the Weierstrass excess function. Motivated by some earlier work by other people, in [23], Morrey has outlined how to extend Weierstrass’s ideas to the higher-dimensional case n > 1 and N = 1. In particular, he has proved the existence of a eld of extremals under the hypotheses in the theorem and has presented the appropriate divergence free structure for the path-independent integral. Using a completely di¬erent technique in [16], Hestenes gave an indirect proof (proof by contradiction) for the case n = N = 1 and later extended this to the case n > 1 and N = 1 (see [17]). The main ingredients in his proof are the concept of directional convergence and certain integral inequalities developed by McShane and later by Reid [24]. It is part of our aim to present an updated version of Hestenes’s argument as rstly it largely motivates the proof of the main results in this paper, and secondly because it seems to be quite unknown to the researchers in the eld. Let us x N = 1 and recall the su¯ cient conditions to be satis ed by the stationary point u0 2 C 1 (« · ) to be a strong local minimizer for I in A1u0 (@« ) when f is of class C 2 . (a) The pointwise positivity of the second variation. For all non-zero ’ 2 W01;2 (« ), ¯ 2 I(u0 ; ’) > 0. (b) The strengthened condition of Legendre. There exists ® > 0 such that fpi pj (x; u0 (x); ru0 (x))¶ i ¶
j
> ® j¶ j2
for all x 2 « · and all ¶ 2 Rn . (c) The strengthened condition of Weierstrass. There exists " > 0 such that Ef (x; u; p; q) := f (x; u; q) for all x 2 « · , ju
u0 (x)j < ", jp
f (x; u; p)
fpi (x; u; p)(qi
pi ) > 0
ru0 (x)j < " and all q 2 Rn .
The rst two conditions and their relation to condition (ii) introduced earlier are studied in xx 2, 4 and 5 of this paper. The function Ef in the third condition is known as the Weierstrass excess function. It is obvious that f (x; u; ¢) is convex at a point p if and only if Ef (x; u; p; q) > 0 for all q 2 Rn . Thus the last condition is a convexity requirement on f with respect to the gradient argument. It is important to point out that this convexity assumption has a central role in all the arguments
A. Taheri
158
mentioned earlier for the case N = 1. For the general case N > 1, it was shown by Meyers [21] (see also Ball [2]) that if u0 is a C 1 strong local minimizer of I, the function f (x; u0 (x); ¢) is quasiconvex at Du0 (x) for each x 2 « . (The smoothness of u0 can be relaxed (cf. [19,31]).) Recall that a continuous function f : RN £n ! R is quasiconvex at A 2 RN £n (cf. [2, 22]) if and only if Z f (A) 6 f (A + D’) dx Q
for all ’ 2 W01;1 (Q; RN ), where Q » Rn is the unit n-cube. It is known that quasiconvexity is weaker than convexity and coincides with the latter when N = 1. Thus, in the above list, the last condition is a somewhat reasonable strengthened version of the necessary condition just described. However, when N > 1, it is far more stronger than being necessary. A major question in this regard is to formulate a set of su¯ cient conditions for strong local minimizers in the case N > 1 that is based on quasiconvexity. This seems to be an open problem. Let us point out that in almost all the su¯ ciency proofs mentioned earlier the function f is assumed to be smooth (at least of class C 2 ) and the stationary point u0 is assumed to be of class C 1 . Hence a further open problem would be to present su¯ cient conditions for local minimizers of I under less smoothness assumptions. We remark that recently Ball and James (unpublished work) have given a direct proof for the su¯ ciency theorem in the case n = 1, under slightly weaker hypotheses. There it is shown that the conclusion of the theorem follows when the rst condition is replaced by u0 is a weak local minimizer of I: As a further remark in [28] and [29], Sivaloganathan has employed the idea of a divergence free structure to treat the case N > 1 under weaker convexity assumptions but in a special case (cf. also the work of Ball and Murat [4]). Another interesting issue is to formulate su¯ cient conditions for u0 to be an Lr local minimizer of I when r < 1. In this general format this appears to be a di¯ cult question. The example Z jruj2 (1 u2 ) dx; I(u) = «
with u0 = 0, shows that the above su¯ cient conditions in general would not give rise to such local minima. Indeed, here u0 = 0 is a strong local minimizer of I in A10 (@« 1 ) but not an Lr local minimizer for any r < 1. To show this let us take a non-zero ’ 2 C01 (Rn ) with supp ’ » B. Here, C01 (Rn ) stands for the class of smooth functions with compact support in Rn . Let us also assume that B » « and consider the sequence ’" (x) := " ¬ ’(x=") for " ! 0+ and some ¬ > 0 to be speci ed later. Clearly, ’" ! 0 in Lr (« ) if ¬ < n=r. Moreover, Z jr’(" 1 x)j2 (1 " 2¬ j’(" 1 x)j2 ) dx I(’" ) = " 2(¬ + 1) « ³Z ´ Z n 2(¬ + 1) 2 2¬ 2 2 jr’(x)j dx " jr’(x)j j’(x)j dx : =" «
«
Su± ciency theorems for local minimizers
159
Figure 1. The double-well potential F .
Hence, if we choose ’ such that the second integral on the right is non-zero, it follows that for any 1 6 r < 1 we can nd 0 < ¬ < n=r such that ’" ! 0 in Lr (« ) while I(’" ) < I(0) for " su¯ ciently small. Theorem 3.3 provides to some extent an answer to the question raised above. It is a generalization of an earlier work by the author in [30] for functionals of the form (1.1) with f (x; u; Du) = jDuj2 + F (x; u). We remark that in this latter case Brezis and Nirenberg [9] have recently established conditions for a weak local minimizer of I to be a W 1;2 local minimizer. Our results here, as well as in [30], are in the somewhat same direction as theirs but obviously in the more general context of Lr (and W 1;r ) local minimizers. Indeed, we here show that the growth of the second derivatives of f , namely, fuu and fup , determine the exponent 1 6 r 6 1 and we present an exact expression for this dependence. We shall also see that, unlike the case in, for example, [9], our proof does not rely on N = 1 and hence the result is valid for N > 1. As a simple example let us consider the case f (x; u; Du) = jDuj2 + F (u), where F 2 C 2 (RN ; R) is a usual double-well potential with two local minima occurring at u = a and u = b (cf. gure 1). As F is bounded from below here, F 2 coincides with the Sobolev space W 1;2 (« ; RN ). It is obvious that u2 = b is the global minimum of I over F 2 . We would, however, like to know about the stationary point u1 = a. According to theorem 3.3, u1 is an L1 local minimizer of I in A2u1 (;) (which is clearly not a global minimizer). This is surprisingly independent of how deep the second well is, i.e. how large the quantity F (a) F (b) might get. To check this, we only need to verify condition (ii) of the theorem, as condition (i) is satis ed by any stationary point of F . But Z 2 ¯ I(u1 ; ’) = (2jD’j2 + Fui uj (a)’i ’j ) dx > ® k’k2W 1;2 (« ;RN ) «
1;2
for all ’ 2 W (« ; R ) and for some ® > 0, provided the Hessian of F at a is strictly positive de nite. Let us end this introduction by describing brie®y the plan of this paper. In x 2 we present some of the important necessary conditions satis ed by various kinds of local minimizers. These are mainly of second order and hence both the function f and the stationary point u0 are assumed to have the required degrees of smoothness. In addition, we mention the appropriate strengthened version of these conditions and also derive some of their basic consequences. In x 3 we state the su¯ ciency N
A. Taheri
160
theorems 3.1 and 3.3 but postpone their proofs to the end of x 4. We begin x 4 by proving some auxiliary results on weak convergence and lower semicontinuity of variational integrals. We also study the question of positivity for quadratic forms and its relation to some of the necessary conditions stated in x 2. Finally, in x 5, as a simple example, we show how the local stability result of Sivaloganathan [29] can be achieved without any need for the construction of local elds and Hamilton{Jacobi theory. For other applications of our results, we refer the reader to [6]. 2. Preliminaries We start this section by discussing the necessary conditions satis ed by di¬erent kinds of local minimizers of the functional (1.1) and mention the appropriate strengthened version of these conditions suitable for the su¯ ciency theorems appearing in the subsequent sections. We shall assume N = 1. Recall that « » Rn is a bounded domain with Lipschitz boundary @« . Moreover, @« = @« 1 [ @« 2 [ N , where @« 1 and @« 2 are disjoint, relatively open and Hn 1 (N ) = 0. Proposition 2.1 (The necessary condition of Legendre). Let f 2 C 2 (« · £ R £ Rn ; R) and let u0 2 C 1 (« · ) be a weak local minimizer of I. Then, for every x 2 « · and for all ¶ 2 Rn , fpi pj (x; u0 (x); ru0 (x))¶ i ¶ j > 0: (L) It is worth noting that condition (L) is actually a consequence of ¯ 2 I(u0 ; ’) > 0 for ’ 2 W01;2 (« ), which itself is a necessary condition to be satis ed by any weak local minimizer of class C 1 . The proof of proposition 2.1 now follows by applying proposition 2.3 to the functional J (’) = ¯ 2 I(u0 ; ’) with ’0 = 0 being a global minimizer. Definition 2.2. The function u0 2 C 1 (« · ) is said to satisfy the strengthened condition of Legendre if and only if there exist ® > 0 such that fpi pj (x; u0 (x); ru0 (x))¶ i ¶
j
> ® j¶ j2
(L+ )
for all x 2 « · and all ¶ 2 Rn . The Weierstrass excess function Ef corresponding to f was de ned earlier in x 1. The following condition restricts the values of ru0 (x) for any strong local minimizer to the set where f (x; u0 (x); ¢) is convex. Proposition 2.3 (The necessary condition of Weierstrass). If u0 2 C 1 (« · ) is a strong local minimizer of I then Ef (x; u0 (x); ru0 (x); q) > 0
(W)
for every x 2 « · and for all q 2 Rn . The proof of this proposition is well known and can be found in [16]. Definition 2.4. The function u0 2 C 1 (« · ) is said to satisfy the strengthened condition of Weierstrass if and only if there exists " > 0 such that Ef (x; u; p; q) > 0
(W+ )
Su± ciency theorems for local minimizers
161
for all x 2 « · , for all u 2 R with ju u0 (x)j < ", for all p 2 R with jp ru0 (x)j < " and for all q 2 Rn . n
n
In the study of su¯ ciency theorems for strong local minimizers of I, essential use is made of the L-function de ned by L(t) := (1 + t2 )1=2
for t 2 R:
1
This function was rst introduced by McShane and later used by various authors, in particular, Reid [24] and Hestenes [16, 17] (see also Bliss [7]). It can easily be checked that L is convex and satis es ) (i) L(t) 6 jtj; (2.1) (ii) L(t) 6 12 t2 : Further properties of this function are stated in the following result. Proposition 2.5 (McShane, Reid [24]). Let L be as above and let ¬ > 0. Then (i)
¬ min(1; ¬ )L(t) 6 L(¬ t) 6 ¬ max(1; ¬ )L(t);
(ii)
t min(¬ ; t) 6 ((1 + ¬
2 1=2
)
+ 1)L(t):
The fact that the L-function is quadratic near the origin and grows linearly at in nity makes it a favourable candidate for acting as a lower bound on the growth of the Weierstrass excess function. This is stated more clearly in the following. Proposition 2.6 (McShane, Reid [24], Hestenes [16, 17]). Let f 2 C 2 (« · £ R £ Rn ; R) and u0 2 C 1 (« · ) satisfy (L+ ) and (W+ ). Then there exist ¬ , " > 0 such that Ef (x; u; p; q) > ¬ L(jq pj) (2.2) n for all x 2 « · , ju u0 (x)j < ", jp ru0 (x)j < " and all q 2 R . The following property of the L-function was suggested and proved by McShane for the case n = 1 and later reformulated and used by Reid (cf. [24]). It was extended to n > 1 and N = 1 by Hestenes (cf. [17]). Here we consider the case where both n; N > 1. Proposition 2.7. Given ¯ > 0, there exist C1 , C2 > 0 such that Z (i) (L(jDuj) C1 juj2 ) dx > 0; « Z (ii) (L(jDuj) C2 jujjDuj) dx > 0; «
for all u 2
W01;1 («
; R ) satisfying kukL1 N
(« ;RN )
6¯ .
Proof. Let a; b 2 Rn be such that (a; b) := (a1 ; b1 ) £ ¢ ¢ ¢ £ (an ; bn ) ¼ « . Given 1 6 i 6 n, let (x0 ; t) denote the n-tuple (x1 ; : : : ; xi = t; : : : ; xn ) and set u = 0 in (a; b)n« · . It follows now (cf. [13, p. 164]) that the function u(x0 ; ¢) 2 W01;1 ((ai ; bi ); RN ) for L n 1 -a.e. x0 . Now, for xed i and x0 , we de ne Z t ju;i (x0 ; z)j dz: v(t) := ai
A. Taheri
162 ju(x0
Clearly, ; t)j 6 v(t) for all t 2 [ai ; bi ] and v 0 (t) = ju;i (x0 ; t)j for a.e. t 2 (ai ; bi ). It follows from Jensen’s inequality for convex functions and proposition 2.5 (i) that Z bi L(ju;i (x0 ; t)j) dt > L((bi ai ) 1 v(bi )) ai
> (bi Moreover, Z
bi
Z
bi
ju(x0 ; t)jju;i (x0 ; t)j dt 6
ai
1
ai )
min(1; (bi
1
ai )
)L(v(bi )):
min(¯ ; v(t))v0 (t) dt
ai
³Z 6 min
bi
Z 0
¯ v (t) dt; ai
´
bi
0
v(t)v (t) dt ai
6 12 v(bi ) min(2¯ ; v(bi )) 6 12 ((1 + 4¯ 2 )1=2 + 1)L(v(bi )); where in the last inequality we have used proposition 2.5 (ii). We can therefore deduce that Z bi Z bi 0 L(ju;i (x ; t)j) dt > Cju(x0 ; t)jju;i (x0 ; t)j dt (2.3) ai
ai
for some C > 0 independent of i. As Z 0 2 ju(x ; t)j 6 2
bi
ju(x0 ; t)jju;i (x0 ; t)j dt;
ai
it follows from (2.3) that Z bi
(L(ju;i (x0 ; t)j)
C1 ju(x0 ; t)j2 ) dt > 0:
ai
A further integration of (2.3) and the above inequality gives Z (L(ju;i j) Cjujju;i j) dx > 0 «
and Z C1 juj2 dx) > 0:
(L(ju;i j) «
The result follows by recalling that L(jDuj) > L(ju;i j) for a.e. x 2 « C2 = C=n.
and setting
Corresponding to the L-function introduced earlier, we can assign the functional R : W01;1 (« ) ! [0; 1); where
Z R(u) =
L(jruj) dx: «
Su± ciency theorems for local minimizers
163
Note that this functional is non-homogeneous and non-subadditive. Moreover, when u0 is su¯ ciently smooth, R(u0 ) represents the di¬erence between the `area’ of the hypersurfaces corresponding to u = u0 and u = 0. In x 4 we shall discuss further properties of this functional. The following proposition plays an important role in the proof of theorem 3.3. It was rst stated and proved for the case s = 0 in [30]. Proposition 2.8. Let 1 6 q, 0 6 s < q, p > q s and de¯ne ³ ³ ´´ p r := r(n; p; q; s) = max 1; n 1 : q s Then, for any ¶ > 0, there exists " > 0 such that Z J (u) := (jDujq + jujq ¶ jDujs jujp ) dx > 12 kukqW 1;q («
;RN )
«
for all u 2 W 1;q (« ; RN ) satisfying kukLr («
;RN )
< ".
Proof. First note that Z J (u) > kukqW 1;q («
((jDujs + jujs )jujp ) dx: ¶
;RN )
«
Moreover, ³Z
Z s
s
p
(jDuj + juj )juj dx 6 «
C1 kuksW 1;q (« ;RN )
juj
p(q=(q s))
´(q dx
s)=q
«
We shall now consider three distinct cases. Case 1 (1 6 q < n). Then Z Z jujp(q=(q s)) dx = jujq jujq(p=(q «
«
³Z 6
juj
q¤
s) 1)
dx
´q=q ¤ ³Z
´q=n n(p=(q s) 1)
juj
dx
«
dx
«
and ³Z ¤
´q=q¤ 6 CkukqW 1;q («
jujq dx «
;RN )
:
Therefore, ³ J (u) > kukqW 1;q (« > provided kukLr («
;RN )
³Z 1
1 kukqW 1;q (« ;RN ) ; 2 ;RN )
´(q jujn(p=(q
¶ C2
is su¯ ciently small.
«
s) 1)
dx
s)=n ´
:
A. Taheri
164
Case 2 (2 6 n 6 q). Setting t = nq=(n + q), it can be checked that t¤ = q and 1 6 12 n 6 t < n for the given range of q. Thus Z Z p(q=(q s)) juj dx = (jujp=(q s) )q dx «
«
³Z p=(q s)t
6C
(juj
+ juj
(p=(q s) 1)t
´q=t jDuj ) dx ; t
«
where we have applied the embedding W 1;t (« ) ,! Lq (« ) to the function jujp=(q (note that p > q s). Using Holder’s inequality, we can now write Z Z jujp=(q s)t dx = jujnq=(n+ q)(p=(q s) 1) jujnq=(n+ q) dx «
«
³Z
´q=(n+ n(p=(q s) 1)
6
juj
q) ³Z
´n=(n+
q)
q
juj dx
dx
«
s)
:
«
Similarly, ³Z
Z juj
(p=(q s) 1)t
´q=(n+ n(p=(q s) 1)
t
jDuj dx 6
juj
«
q) ³Z
´n=(n+ q
jDuj dx
dx
«
q)
:
«
Therefore, ³Z
Z jujp(q=(q
s))
«
dx 6 CkukqW 1;q («
´q=n jujn(p=(q
;RN )
s) 1)
dx
;
«
and so the result follows similar to that in case 1. Case 3 (n = 1). Without loss of generality, let « Z 1 Z 1 jujp(q=(q s)) dx = juj(p=(q s) 1) juj1+ 0
= (0; 1). We can now write (p=(q s))(q 1)
0
Z
6 kjuj(1+
(p=(q s))(q 1))
kL1
1
(0;1)
dx
juj(p=(q
s) 1)
dx:
0
Applying the embedding W 1;1 (0; 1) ,! L1 (0; 1) to the function juj1+ and using a Holder inequality, we have Z
³Z
1 p(q=(q s))
juj 0
´
1
dx 6 C
(p=(q s))(q 1)
(p=(q s) 1)
juj 0
³Z
dx
1
jujq(p=(q
£ 0
s))
´(q dx
1)=q ³Z 1
´1=q q
(juj + juxjq ) dx
;
0
and so the result follows immediately. It is possible to show that the exponent r de ned in proposition 2.8 is sharp. We shall refer the interested reader to [30] for a proof of this. We end this section with the following.
Su± ciency theorems for local minimizers Example 2.9. Let p1 ; p2 > 0 and ¶
1; ¶ 2
165
> 0 be given. Then setting
r = max(1; 12 np1 ; np2 ); there exist " > 0 such that Z (jDuj2 + juj2 ¶ 1 jujp1 +
2
¶
p2 + 1 ) dx 2 jDujjuj
«
for all u 2 W 1;2 (« ; RN ) satisfying kukLr («
> 12 kuk2W 1;2 («
;RN )
< ".
;RN )
3. Statement of the su± ciency theorems In this section we state the su¯ ciency theorems for local minimizers of I. The proofs are given in x 4. Recall that corresponding to the functional (1.1) we assign the class of admissible functions F q as in (1.2) and, for a xed u0 2 F q and @« 1 » @« , we set Aqu0 (@« 1 ) = fu 2 F q : (u u0 )j@« 1 = 0g: We can now state the following. Theorem 3.1 (The fundamental su¯ ciency theorem). Let « » Rn be as described earlier and consider the functional (1.1) with f 2 C 2 (« · £ R £ Rn ; R). Let u0 2 C 1 (« · ) satisfy (W + ) and (i)
¯ I(u0 ; ’) = 0; ¯ 2 I(u0 ; ’) > ® k’k2W 1;2 («
(ii)
)
for some ® > 0 and all ’ 2 W 01;2 (« ). Then there exist ¼ , » > 0 such that I(u0 ) > ¼ R(u
I(u) for all u 2 A1u0 (@« ) satisfying ku
u 0 k L1
(« )
u0 )
® k’k2W 1;2 («
(ii)
for some ® > 0 and all ’ 2 W 1;2 (« ) with ’j@«
1
)
= 0.
(1) (The case r = 1.) Assume that there exist ¬ , " > 0 such that Ef (x; u; ru0 (x); q) > ¬ jq for all x 2 « · , ju
(3.1)
u0 (x)j < " and q 2 Rn . Then there exist ¼ , » > 0 such that I(u)
for all u 2 A2u0 (@«
ru0 (x)j2
1 ),
I(u0 ) > ¼ ku
provided ku
u0 kL1
u0 k2W 1;2 (« (« )
0 such that Ef (x; u; ru0 (x); q) > ¬ jq
ru0 (x)j2
(3.3)
for all x 2 « · , u 2 R and q 2 Rn . Furthermore, let, for some p1 , p2 > 0, jfuu (x; u; ru0 (x))j 6 C(1 + jujp1 )
jfup (x; u; ru0 (x))j 6 C(1 + jujp2 ) (3.4) for some C > 0 and all x 2 « · . Then there exist ¼ , » > 0 such that (3.2) holds for all u 2 A2u0 (@« 1 ), provided ku u0 kLr (« ) < » , where and
r = r(n; p1 ; p2 ) = max(1; 12 np1 ; np2 ): Note that in the above theorem (case 1), the lower bound on I(u) I(u0 ) is sharper than that of theorem 3.1, as kr(u u0 )k2L2 (« ;Rn ) > 2R(u u0 ). We have achieved this by imposing a quadratic growth on f with respect to the gradient at in nity (cf. (3.1)). We also remark that in the special case f (x; u; ru) = jruj2 + F (x; u), condition (3.3) trivially holds with ¬ = 1 and that r = r(n; p1 ; p2 ) = max(1; 12 np1 ). In this way, we recover the results in [30] (cf. also [9]). 4. Proofs This section is devoted to the proof of theorems 3.1 and 3.3. We rst discuss some auxiliary results and postpone the proofs of these theorems to x 4.5. 4.1. Lower semicontinuity of quadratic forms (k) Let faij g be a given sequence of measurable functions on « and such that ! aij (the mode of convergence to be speci ed later). In this subsection we study the question of lower semicontinuity in the following setting Z Z (k) (k) (k) aij (x)’;i ’;j dx 6 lim inf aij (x)’;i ’;j dx (k) aij
k!
«
1
«
(k) aij (x)
(k)
(k)
when ’ * ’ in W (« ). As replacing with (aij (x) + 12 aji (x)) does not change the integrands, we can assume without loss of generality that the sequence fa(k) ij g is symmetric, that is, (k)
1;2
(k)
(k)
aij (x) = aji (x)
for a.e. x 2 «
and for all 1 6 i, j 6 n.
We shall start by recalling two well-known results. Lemma 4.1 (A re nement of Fatou’s lemma). Let fu(k) g be a sequence of measurable functions such that f(u(k) ) g is uniformly integrable and u(k) ! u a.e. Then Z Z u dx 6 lim inf u(k) dx: «
k!
1
«
The proof of this lemma is an easy exercise in measure theory and we refer the reader to [18] and [25]. We also need the following result.
Su± ciency theorems for local minimizers 1
167
Lemma 4.2. Let a 2 L (« ; R ) satisfy the following version of Legendre’s condition aij (x)¶ i ¶ j > 0 for a.e. x 2 « (4.1) n£n
for all ¶ 2 Rn . Then the variational integral Z J (’) = aij (x)’;i ’;j dx «
is sequentially weakly lower semicontinuous on W 1;2 (« ). An almost immediate consequence of this is the following. Proposition 4.3. Let a(k) ! a in L1 (« ; Rn£n ) with (aij ) satisfying (4.1). Then Z Z (k) (k) (k) aij (x)’;i ’;j dx 6 lim inf aij (x)’;i ’;j dx 1
k!
«
«
when ’(k) * ’ in W 1;2 (« ). Proof. We claim that
Z
lim inf
Z
1
k!
«
(k) (k) (k) aij ’;i ’;j
(k)
dx = lim inf 1
k!
«
Indeed, this can be seen from Z Z Z (k) (k) (k) (k) (k) a ’ ’ dx 6 ja(k) aij ’;i ’;j dx ij ;i ;j «
«
(k)
aij ’;i ’;j dx:
ajjr’(k) j2 dx
«
6 ka(k)
akL1
(« ;Rn£
n
) k’
(k) 2 kW 1;2 (« ) :
An application of lemma 4.2 now completes the proof. We can now make the following general statement concerning the lower semicontinuity question raised at the beginning of this section. (k) Proposition 4.4. Let faij g be a sequence of measurable functions such that (k) aij ! aij a.e. and let ’(k) * ’ in W 1;2 (« ). Furthermore, assume (aij ) satis¯ es (k) (k) (k) (4.1) and that the sequence f(aij ’;i ’;j ) g is uniformly integrable. Then Z Z (k) (k) (k) aij (x)’;i ’;j dx 6 lim inf aij (x)’;i ’;j dx: k!
«
1
«
Remark 4.5. The hypotheses in proposition 4.4 are weaker than those of lemma 4.1 in the sense that the weak convergence of fr’(k) g in L2 (« ; Rn ) does not imply any kind of pointwise convergence. This proposition exhibits how convexity can `handle’ weak convergence in the context of lower semicontinuity. Proof. An application of Egoro¬’s theorem to the sequence fa(k) g shows that a(k) ! a almost uniformly in « . This means that, for a sequence f« (l) g of measurable subsets of « shrinking to zero, i.e. « (l+ 1) » « (l) and L n (« (l) ) ! 0, a(k) ! a
L1 (« n«
(l)
; Rn£n )
for all l:
A. Taheri
168
By the uniform integrability condition, given " > 0, there exists ¯ > 0 such that Z (k) (k) (k) (aij ’;i ’;j ) dx 6 " whenever L n (E) < ¯ : E
Therefore, for l su¯ ciently large, Z Z (k) (k) (k) (k) (k) (k) aij ’;i ’;j dx > aij ’;i ’;j dx «
Z
« n«
(l)
« n«
(l)
(k)
>
(k)
(k)
aij ’;i ’;j dx
Z (k)
«
(l)
(k)
(k)
(aij ’;i ’;j ) dx
":
Letting k ! 1 and applying the previous proposition to the rst term, we obtain Z Z (k) (k) (k) lim inf aij ’;i ’;j dx > aij ’;i ’;j dx ": 1
k!
« n«
«
(l)
The result now follows by letting l ! 1 and an application of Lebesgue’s theorem on monotone convergence. By slightly modifying the above proof we can also show the following. (k)
Proposition 4.6. Let faij g be a sequence of measurable functions such that (k) aij ! aij almost uniformly and let r’(k) * r’ in L2 (« A ; Rn ) for each measur(k) able « A » « on which aij ! aij uniformly. Furthermore, assume (aij ) satis¯ es (k) (k) (k) (4.1) and that f(aij ’;i ’;j ) g is uniformly integrable. Then Z Z (k) (k) (k) aij (x)’;i ’;j dx 6 lim inf aij (x)’;i ’;j dx: 1
k!
«
«
We now turn our attention to quadratic functionals of the form Z J (’) = (aij (x)’;i ’;j + bi (x)’;i ’ + c(x)’2 ) dx; «
where the coe¯ cients a, b and c are all bounded measurable functions and where (aij ) satis es the following version of the strengthened Legendre condition, namely there exists ® > 0 such that aij (x)¶ i ¶ for a.e. x 2 «
j
> ® j¶ j2
and all ¶ 2 Rn .
Proposition 4.7. Let J be as above and assume J (’) > 0 for all non-zero ’ 2 W 1;2 (« ) satisfying ’j@« 1 = 0. Then there exists ¶ 1 > 0 such that J (’) > ¶ for all ’ 2 W 1;2 (« ) with ’j@«
1
= 0.
2 1 k’kW 1;2 (« )
Su± ciency theorems for local minimizers
169
Recalling that the second variation of I at any point u0 2 C (« · ) is a quadratic functional of the above type, more speci cally, Z ¯ 2 I(u0 ; ’) = (fpi pj (x; u0 ; ru0 )’;i ’;j 1
«
+ 2fpi u (x; u0 ; ru0 )’;i ’ + fuu (x; u0 ; ru0 )’2 ) dx;
we can state the following. Corollary 4.8. Let I, « , u0 be as in theorem 3.1. Then condition (ii) is equivalent to (L+ ) and ¯ 2 I(u0 ; ’) > 0 for all non-zero ’ 2 W 1;2 (« ) satisfying ’j@«
1
= 0.
Remark 4.9. This result is still true when N > 1 if (L+ ) is replaced by the strong ellipticity condition (cf. x 5) and provided @« 1 = @« . For the case @« 2 6= ;, an additional condition known as the complementing condition should hold for every x 2 @« 2 (cf. [26]). Proof of proposition 4.7. Step 1. By a standard minimization of J over the unit sphere in L2 (« ), it follows that there exists ¬ > 0 such that J (’) > ¬ k’k2L2 (« for all ’ 2 W 1;2 (« ) satisfying ’j@«
1
= 0, so that 1 ¬ 2
J1 (’) := J (’)
(4.2)
)
k’k2L2 («
)
> 12 ¬ k’k2L2 («
):
Step 2. We now claim that J1 (’) > kr’k2L2 («
;Rn )
for some > 0. Indeed, if this were not the case, there would be a sequence of non-zero functions f’(k) g such that 1 kr’(k) k2L2 (« k
;Rn )
> J1 (’(k) ) > 12 ¬ k’(k) k2L2 («
)
> 0:
Note that from this it follows that kr’(k) kL2 («
;Rn )
6= 0;
and so letting ã (k) = ’(k) =kr’(k) kL2 (« ;Rn ) and appealing to the quadratic nature of J1 , we get Z 1 (k) (k) (k) > (aij ã ;i ã ;j + bi ã ;i ã (k) + (c 12 ¬ )(ã (k) )2 ) dx > 12 ¬ kã (k) k2L2 (« ) : (4.3) k « The boundedness of the sequence fã (k) g in W 1;2 (« ) implies that, by passing to a subsequence, ã (k) * ã in W 1;2 (« ); ã (k) ! ã in L2 (« );
A. Taheri
170
and so ã = 0. By passing to the limit in (4.3) as k ! 1, Z Z (k) (k) ã ã 0 > lim inf aij ;i ;j dx + (bi ã ;i ã + (c k!
1
«
«
1 2¬
)ã 2 ) dx:
This, together with the fact that Z «
(k) (k) aij ã ;i ã ;j dx > ® ;
contradicts ã = 0. The proof is thus complete. A direct proof for proposition 4.7. By step 1 of the previous proof, we can assume equation (4.2). Clearly, for any "1 > 0, we have Z Z 2 ((1 + "1 )aij ’;i ’;j + bi ’;i ’ + c’ ) dx > ("1 aij ’;i ’;j + ¬ ’2 ) dx; «
or alternatively, Z ³ aij ’;i ’;j + «
«
´ 1 "1 ® 2 kr’k2L2 (« (bi ’;i ’ + c’ ) dx > 1 + "1 1 + "1
;Rn ) +
Now we wish to prove ´ Z ³ 1+¼ aij ’;i ’;j + (bi ’;i ’ + c’ 2 ) dx > k’k2L2 (« 1 + "1 «
¬ k’k2L2 (« 1 + "1
):
)
for some > 0 and ¼ > "1 . Then this would imply the desired uniform positivity in the W 1;2 sense. However, the left-hand side of the above inequality can be written as ´ Z ³ 1 ¼ 2 2 aij ’;i ’;j + (bi ’;i ’ + c’ ) + (bi ’;i ’ + c’ ) dx 1 + "1 1 + "1 « "1 ® ¬ > kr’k2L2 (« ;Rn ) + k’k2L2 (« ) 1 + "1 1 + "1 M¼ (kr’kL2 (« ;Rn ) k’kL2 (« ) + k’k2L2 (« ) ) 1 + "1 1 2 1 > (("1 ® ¼ M=2s M ¼ )k’k2L2 (« ) ); 2 ¼ M s)kr’kL2 (« ;Rn ) + (¬ 1 + "1 with max(kbkL1
(« ;Rn ) ; kckL1 (« ) )
"1 ®
1 ¼ 2
6 M and s > 0 arbitrary. Therefore, we require
M s > 0;
¬
¼ M (1=2s + 1) > 0;
plus the fact that "1 < ¼ . These can be written as 1
1=p1 + 1=p2 , with the usual interpretations for 1.
Note that in the particular case where p1 and p2 are conjugate exponents, the product sequence converges weakly in L1 (« ) to the product of the limits. Proposition 4.11. Assume that n > 3 and let f’(k) g be a bounded sequence in W 1;2 (« ). Then, by passing to a subsequence if necessary, ’(k) * ’
(i)
(’(k) )2 * ’2
(ii)
in W 1;2 (« ); ¤
in W 1;1 (« ):
Furthermore, if fb(k) g and fc(k) g are sequences such that b(k) ! b in Ln (« ; Rn ) and c(k) ! c in Ln=2 (« ), then ) (iii) c(k) (’(k) )2 * c’2 in L1 (« ); (4.5) (k) (k) (iv) bi ’(k) ’;i * bi ’’ ;i in L1 (« ): Proof of proposition 4.11. As a result of the re®exivity of W 1;2 (« ) and the compactness of the embedding W 1;2 (« ) ,! Lp (« ) for p < 2¤ , it follows that, by passing to a subsequence if necessary (we do not re-label this), we have ’(k) * ’
in W 1;2 (« );
’(k) ! ’
a.e. in « :
Now let ã (k) = (’(k) )2 . Then rã (k) = 2’(k) r’(k) a.e. and therefore, by a simple application of Holder’s inequality, we have that, since 1¤ =2¤ + 1¤ =2 = 1, ³Z ´1¤ =2¤ ³Z ´1¤ =2 Z ¤ ¤ (k) 1 (k) 2 (k) 2 jrã j dx 6 C0 j’ j dx jr’ j dx < C1 : «
Also,
«
«
Z
Z ¤
¤
jã (k) j1 dx = «
j’(k) j1
:2
dx < C2 ;
«
as 1¤ :2 < 2¤ . Therefore, fã (k) g is bounded in W 1;1 (« ) and so, by passing to a ¤ further subsequence, ã (k) * À for some À in W 1;1 (« ). The pointwise convergence ã (k) ! ’2 now implies À = ’2 a.e. ¤¤ To show the next part we rst recall that (’(k) )2 * ’2 in L1 (« ) = Ln=(n 2) (« ) ¤ as a result of the continuity of the embedding W 1;p (« ) ,! Lp (« ). An application of lemma 4.10 with p1 = 12 n, p2 = n=(n 2) and r = 1 now implies the weak convergence c(k) (’(k) )2 * c’2 . The other case is similar. ¤
A. Taheri
172
Proposition 4.12 (Hestenes [17]). Let f’(k) g be a sequence in W01;1 (« ) such that Z 9 > > j’(k) j2 dx < 1; (i) sup = k « Z (4.6) > ; j’(k) jjr’(k) j dx < 1> (ii) sup k «
for some ’ 2 W01;2 (« ). Then, for any b 2 C(« · ; Rn ), Z Z (k) (k) lim bi (x)’ ’;i dx = bi (x)’’;i dx:
and ’(k) ! ’ a.e. in «
1
k!
«
«
It is important to note that in this proposition the restriction on the functions ’(k) to vanish on the boundary is essential. Proof. We shall give this in two steps. Step 1. We prove the result for the case when b 2 C 1 (« · ; Rn ). It follows from (4.6) that the sequence f(’(k) )2 g is bounded in W 01;1 (« ). (Note that (4.6) implies that, for each k, the weak derivative r(’(k) )2 = 2’(k) r’(k) (cf. [14, p. 151]).) Thus it follows from the compactness of the embedding W 1;1 (« ) ,! L1 (« ) that, by passing to a subsequence if necessary, (’(k) )2 ! ’2 in L1 (« ). Now (k)
bi ’(k) ’;i =
1 @ (bi (’(k) )2 ) 2 @xi
1 b (’(k) )2 ; 2 i;i
and hence an application of the divergence theorem shows that Z Z 1 (k) bi ’(k) ’;i dx = bi;i (’(k) )2 dx: 2
«
«
This implies the conclusion. Step 2. We now consider the general case when b 2 C(« · ; Rn ). By approximation, it follows that, for any given " > 0, there exists b¤ 2 C 1 (« · ; Rn ) such that kb b¤ kL1 (« ;Rn ) < ". Therefore, Z bi (’(k) ’(k) ’’;i ) dx ;i « Z Z ¤ ¤ (k) (k) (k) (k) 6 jbi bi jj’ ’;i ’’;i j dx + bi (’ ’;i ’’;i ) dx « « Z (k) 6 "C + b¤i (’(k) ’;i ’’ ;i ) dx «
6 "(C + 1); provided k is su¯ ciently large (we have used the result from step 1 for the second integral). The proof is complete since " is arbitrary. Corollary 4.13. Let f’(k) g be as above and let fb(k) g be a sequence such that b(k) ! b in L1 (« ; Rn ) with b 2 C(« · ; Rn ). Then Z Z (k) (k) lim bi (x)’(k) ’;i dx = bi (x)’’;i dx: k!
1
«
«
Su± ciency theorems for local minimizers Furthermore, if c
(k)
173
n
! c in L (« ), then c(k) (’(k) )2 * c’2
in L1 (« ):
(Compare with (4.5).) Proof. The rst part follows by noting that Z Z (k) (k) (k) lim bi ’(k) ’;i dx = lim bi ’(k) ’;i dx: 1
k!
1
k!
«
Indeed, Z (k) (k) (k) b ’ ’ dx ;i i
Z (k) 6 jb(k) bi ’(k) ’;i dx
Z
«
«
«
bjj’(k) jjr’(k) j dx
«
6 Ckb(k)
bkL1
(« ;Rn ) :
For the second part, assume n > 1. Then it follows that the sequence f(’(k) )2 g is ¤ ¤ bounded in L1 (« ) and thus (’(k) )2 * ’2 in L1 (« ). (Note the pointwise convergence given in the proposition.) The result is now a consequence of lemma 4.10. The case n = 1 is similar. 4.3. Some convergence properties related to R The positive functional R was de ned earlier in x 2. The fact that it is nonhomogeneous and non-subadditive makes it far from being a norm over W01;1 (« ); however, it has some features similar to that of the norm k ¢ kW 1;1 (« ) . We start by showing that they have the same convergent sequences. (We note that the results in this subsection are due to Hestenes [17] and are presented in a shorter and updated form for the convenience of the reader.) For a sequence fv(k) g in W01;1 (« ), the convergence R(v (k) ) ! 0 implies rv (k) ! 0 in measure. Recall that a sequence of measurable functions fg (k) g converge to zero in measure if and only if for any " > 0, L
n
(fx 2 « ; jg (k) (x)j > "g) ! 0
as k ! 1:
In fact, one can say more. Proposition 4.14. kv (k) kW 1;1 («
)
! 0 if and only if R(v (k) ) ! 0.
Proof. (a) The implication ()) is trivial. (b) To show ((), we consider ³Z ´2 kruk2L1 (« ;Rn ) = jruj dx «
³Z
³ ´ 1=2 ´2 jruj 1 6 1 + 2 L(ru) dx 1 1=2 « (1 + 2 L(ru)) ³Z ´³Z ´ 6 L(ru) dx (2 + L(ru)) dx : «
«
Therefore, an application of Poincar´e’s inequality shows that kuk2W 1;1 («
)
6 CR(u)(1 + R(u))
A. Taheri
174
for some C > 0. The proof is thus complete. Assume now that the sequence of non-zero functions fv (k) g in W 01;1 (« ) is such that R(v (k) ) ! 0 and de ne a corresponding sequence of variations ’(k) =
v (k) : R1=2 (v (k) )
In what follows we shall explore some properties of this sequence. Proposition 4.15. The sequence f’(k) g is weakly relatively compact in W01;1 (« ). Proof. The result follows by showing that fr’(k) g is weakly relatively compact in L1 (« ; Rn ). For this, let E be a measurable subset of « , then ³Z ´2 Z Z jr’(k) j2 (k) jr’ j dx 6 dx (1 + 12 L(rv (k) ) dx 1 (k) ) E E 1 + 2 L(rv E 6 2(L
n
(E) + 12 R(v (k) )):
Since R(v (k) ) ! 0, it follows that the sequence fr’(k) g is uniformly integrable and thus, according to the Dunford{Pettis criterion, sequentially weakly relatively compact in L1 (« ; Rn ). As a consequence of the above proposition, we can now assume that there exists ’ 2 W01;1 (« ) such that, by passing to a subsequence (we do not re-label this), ’(k) * ’ in W 1;1 (« ). An application of Egoro¬’s theorem to the sequence frv (k) g implies the existence of a sequence f« (l) g of measurable subsets of « , shrinking to zero, such that rv (k) ! 0 in L1 (« n« (l) ; Rn ) for each l. Using this, we can now improve the weak convergence of the sequence of variations f’(k) g as stated in the following. Proposition 4.16. The sequence fr’(k) g lies in L2 (« n« (l) ; Rn ) for su± ciently large k (depending on l) and the variation ’ belongs to W01;2 (« ). Furthermore, r’(k) * r’ in L2 (« n« (l) ; Rn ) for each l. Proof. Consider the sequence fz (k) g, where z (k) =
r’(k) : (1 + 12 L(rv(k) ))1=2
Then
Z kz (k) k2L2 (« ;Rn )
=
and so, by passing to a subsequence, z Then Z (k) (k) ¢ ¢ r’ (g z g ) dx (l)
« (k)
jr’(k) j2 dx = 2; 1 + 12 L(r(v (k) ) * z in L2 (« ; Rn ). Now let g 2 L1 (« ; Rn ).
« n«
6 kgkL1
(«
;Rn )
kr’(k) kL1 («
;Rn )
® ® 1 ® ® (1 + 1 L(rv(k) ))1=2 2
1 L1
: (« n«
(l) )
Su± ciency theorems for local minimizers
175
The convergence of the last term to zero in the above inequality implies that z = r’ for a.e. x 2 « n« (l) and so for a.e. x 2 « ; therefore, ’ 2 W01;2 (« ). In addition, Z Z 1 (k) (k) 2 jr’ jz (k) j2 (L(rv(k) ))2 dx ! 0 z j dx = « n«
2
(l)
(l)
« n«
and therefore r’(k) * r’
in L2 (« n«
(l)
; Rn ):
This completes the proof. 4.4. A lower semicontinuity property for I Given a 2 C(« · £ R; R), b 2 C 1 (« · £ R; Rn ), consider the functional Z J (u) = (a(x; u) + bi (x; u)u;i ) dx: «
The following is an extension of a lower semicontinuity property of J by Lindeberg and Hestenes [17] to the case where @« 2 6= ;. Lemma 4.17. Let u0 2 C 1 (« · ) satisfy bi (x; u0 (x))¸ i (x) = 0 on @« " > 0, there exists ¯ > 0 such that jJ (u)
J (u0 )j 6 "(1 + R(u
2.
Then, for any
u0 ))
for all u 2 W 1;1 (« ) satisfying ku
u0 kL1
(« )
0 such that to any " > 0 there corresponds a ¯ > 0 satisfying I(u)
I(u0 ) > R(u
u0 ) dx
";
whenever ku
u0 kL1
(« )
0 with g having an in nite integral.) However, if at least one of g or h has a nite integral, the above equality holds. Thus special attention need to be paid on such manipulations specially in x 4.5.
A. Taheri
176 Proof of the corollary. Z I(u)
(f(x; u; ru)
I(u0 ) =
f (x; u0 ; ru0 )) dx
Z« (Ef (x; u; ru0 ; ru) + f (x; u; ru0 )
= «
+ fpi (x; u; ru0 )(u
Z >¬
L(jr(u
f(x; u0 ; ru0 )) dx
u0 );i
u0 )j) dx + J (u)
J (u0 );
«
where
Z (f (x; u; ru0 ) + fpi (x; u; ru0 )(u
J (u) =
u0 );i ) dx;
«
and we have used proposition 2.6. The result now follows from the lemma and recalling the natural boundary condition fpi (x; u0 (x); ru0 (x))¸ i (x) = 0 (when @« 2 6= ;). Proof of the lemma. It can easily be checked that Z J (u) J (u0 ) = (a(x; u) a(x; u0 )+(bi (x; u) bi (x; u0 ))u0;i +bi (x; u)(u u0 );i ) dx: «
But bi (x; u)(u
u0 );i =
@ (bi (x; u0 )(u u0 )) @xi @ bi (x; u0 )(u @xi
u0 ) + (bi (x; u)
bi (x; u0 ))(u
and hence an application of the divergence theorem shows that Z bi (x; u)(u u0 );i dx « Z ³ @ = bi (x; u0 )(u u0 ) + (bi (x; u) bi (x; u0 ))(u @xi «
u0 );i ;
´ u0 );i dx:
Therefore, from the continuity assumption on a and b, it follows that, for ku u0 kL1 (« ) su¯ ciently small, Z jJ (u) J (u0 )j 6 (ja(x; u) a(x; u0 )j + jru0 jjb(x; u) b(x; u0 )j «
+ jbi;i (x; u0 )jju u0 j + jb(x; u) Z " 6 (1 + L(jr(u u0 )j)) dx c « ³ ´ Z 6" 1+ L(jr(u u0 )j) dx ; «
where c = max(1; L
n
(« )).
b(x; u0 )jjr(u
u0 )j) dx
Su± ciency theorems for local minimizers
177
4.5. Proof of theorems 3.1 and 3.3 We start this subsection with the proof of theorem 3.1. Proof of theorem 3.1. Assume the conclusion were false. Then there would exist a sequence fu(k) g » A1u0 (@« ) with u(k) di¬erent from u0 , u(k) ! u0 in L1 (« ), such that 1 I(u(k) ) I(u0 ) < R(u(k) u0 ): (4.7) k It follows from corollary 4.18 that, for any " > 0, R(u(k)
u0 )
"
(aij (x)’;i ’;j + bi (x)’;i ’(k) + c(k) (x)(’(k) )2 ) dx: k «
(4.10)
Setting u = u(k) u0 in proposition 2.7, it follows that the sequence f’(k) g satis es (4.6). By passing to the limit in (4.10) and using corollary 4.13, Z (k)
0 > lim inf k!
1
(k)
(k)
«
aij (x)’;i ’;j dx Z 1 + (2fpi u (x; u0 ; ru0 )’;i ’ + fuu (x; u0 ; ru0 )’2 ) dx: 2
(4.11)
«
Since Z
Z (k)
lim inf k!
1
«
(k)
(k)
aij (x)’;i ’;j dx = lim inf k!
1
«
Ef (x; u(k) ; ru0 ; ru(k) ) >¬ >0 R(u(k) u0 )
6 0. (according to (2.2)), it can immediately be deduced that ’ = Recalling the lower semicontinuity result in proposition 4.6, proposition 4.16 and (k) (k) (k) the fact that aij (x)’;i ’;j > 0, it follows from (4.11) that 0 > 12 ¯ 2 I(u0 ; ’): This however contradicts (ii). The proof is thus complete.
(4.12)
Su± ciency theorems for local minimizers
179
We shall now proceed with the proof of theorem 3.3. The main lines of the proof are similar to that of theorem 3.1 and for this reason we will abbreviate some of the arguments. Proof of theorem 3.3. We argue by contradiction. Indeed, if the conclusion were false, for some sequence fu(k) g » A2u0 (@« 1 ) with u(k) di¬erent from u0 , we would have u(k) ! u0 in Lr (« ) and 1 (k) ku u0 k2W 1;2 (« k We now de ne the sequence of normalized variations I(u(k) )
I(u0 )
0. Then the sequence fu(k) g is bounded in W 1;2 (« ). In the case C = 0, we can extract a subsequence such that u(k) ! u0 in W 1;2 (« ). Proof. It follows from the growth condition (3.4) that there exist C1 , C2 , > ¬ (¬ as in (3.3)) such that f (x; u(k) ; ru0 )
f(x; u0 ; ru0 ) >
C1 (1 + ju(k)
u0 jp1 +
2
) + ju(k)
u0 j2 ;
and similarly, (fp (x; u(k) ; ru0 )
fp (x; u0 ; ru0 )) ¢ r(u(k) >
u0 )
C2 (1 + ju(k)
u0 jp2 +
1
)jr(u(k)
u0 )j
for a.e. x 2 « . Therefore, using the expansion (4.8) and condition (3.3), we can write 1 (k) ku u0 k2W 1;2 (« ) + C k Z > (¬ jr(u(k) u0 )j2 C1 (1 + ju(k) u0 jp1 + 2 ) + ju(k) u0 j2 «
C2 (1 + ju(k)
u0 jp2 +
1
)jr(u(k)
u0 )j
C3 jr(u(k)
u0 )j) dx;
A. Taheri
180 and, as ¬ < , Z C4 >
1 2¬
ku
(k)
u0 k2W 1;2 (« ) «
C1 ju(k) Z
u0 jp1 +
2
dx
C5 (1 + ju(k)
u0 jp2 +
1
jr(u(k)
u0 )j dx:
«
Applying proposition 2.8 (with q = 2) to this inequality (cf. the example following the proposition), we get C4 > 14 ¬ ku(k)
u0 k2W 1;2 («
)
C5 kr(u(k)
u0 )kL1 («
;Rn )
for su¯ ciently large k. Using a Holder inequality on the last expression, it follows immediately that the above can hold only if ku(k) u0 kW 1;2 (« ) is bounded. If C = 0, it follows from the previous part that, by passing to a subsequence if necessary, u(k) ! u0 in L2 (« ) (as a result of the compactness of the embedding W 1;2 (« ) ,! L2 (« )). Using (4.9) and the growth condition (3.4), we can write Z 1 (k) 2 ku k > u0 W 1;2 (« ) (¬ jr(u(k) u0 )j2 C6 (1 + ju(k) u0 jp1 )ju(k) u0 j2 k « C7 (1 + ju(k) or
u0 jp2 )ju(k)
u0 jjr(u(k)
u0 )j) dx
Z 0 > 12 ¬ ku(k) u0 k2W 1;2 (« ) C8 (1 + ju(k) u0 jp1 )ju(k) u0 j2 dx « Z C7 (1 + ju(k) u0 jp2 )ju(k) u0 jjr(u(k) u0 )j dx « Z > ( 14 ¬ jr(u(k) u0 )j2 C8 ju(k) u0 j2 C7 ju(k) u0 jjr(u(k) u0 )j) dx; «
where we have again used proposition 2.8. Setting " = ku(k) u0 kL2 (« ) and t = kr(u(k) 0 > 14 ¬ t2
C7 "t
u0 )kL2 («
;Rn ) ,
we have
C8 " 2 ;
or 0 6 t 6 C9 ". The result follows by letting " ! 0. We can now apply lemma 4.19 to the sequence fu(k) g and deduce that, by passing to a subsequence, u(k) ! u0 in W 1;2 (« ). By passing to a further subsequence, this implies that (k) aij ! 12 fpi pj (¢; u0 (¢); ru0 (¢)) for a.e. x 2 « . We shall now consider two cases. Case A (n > 3). It follows from the convergence u(k) ! u0 in Lr (« ) that u(k) ! u0 in Lnp1 =2 (« ) and Lnp2 (« ). Therefore, the growth conditions (3.4), together with Lebesgue’s theorem on dominated convergence, imply that (k)
bi
! fpi u (¢; u0 (¢); ru0 (¢));
c(k) ! 12 fuu (¢; u0 (¢); ru0 (¢))
in Ln (« ) and Ln=2 (« ), respectively. Applying propositions 4.4 and 4.11 to (4.10), we can now deduce that ¯ 2 I(u0 ; ’) 6 0.
Su± ciency theorems for local minimizers
181
Case B (n 6 2). The main problem is that in this case we can not apply proposition 4.11 to the sequence f’(k) g. However, this di¯ culty can be overcome by recalling the conclusion of lemma 4.19. Indeed, it follows from u(k) ! u0 in W 1;2 (« ), the continuity of the embedding W 1;2 (« ) ,! Lq (« ) for q < 1 (W 1;2 (« ) ,! L1 (« ) when n = 1) and the growth conditions (3.4) that (k)
bi
! fpi u (¢; u0 (¢); ru0 (¢));
c(k) ! 12 fuu (¢; u0 (¢); ru0 (¢))
in Lq (« ) for any q < 1. We can again pass to the limit in (4.10) by an application on lemma 4.10. Thus ¯ 2 I(u0 ; ’) 6 0. 6 0. But The only remaining task (in both cases 1 and 2) now is to exhibit ’ = this follows from (4.10), i.e. Z 1 (k) (k) > ¬ k’(k) k2W 1;2 (« ) + (bi (x)’;i ’(k) + (c(k) (x) ¬ )(’(k) )2 ) dx k « or, after passing to the limit and noting that k’(k) kW 1;2 (« ) = 1, Z 1 0>¬ + (2fpi u (x; u0 ; ru0 )’;i ’ + (fuu (x; u0 ; ru0 ) 2¬ )’2 ) dx; 2
«
which is false if ’ = 0. Remark 4.20. Similar to the case N = 1, we can associate to any given su¯ ciently smooth f : « £ RN £ RN £n ! R, the Weierstrass excess function Ef : « £ RN £ RN £n £ RN£n ! R by setting Ef (x; u; P; Q) := f(x; u; Q)
f(x; u; P )
fP (x; u; P ) ¢ (Q
P ):
The statement of theorem 3.3 (for simplicity here we restrict to case 1, similar comments apply to case 2 with the appropriate modi cations) can now be generalized to the case N > 1 if condition (3.1) is replaced by its multi-dimensional analogue. There exist ¬ ; " > 0 such that Ef (x; u; Du0 (x); Q) > ¬ jDu0 (x)
Qj2
(4.14)
for all x 2 « · , ju u0 (x)j < " and Q 2 R . The proof can be extended to this case without much di¯ culty. The important point, however, is that this condition is much stronger than necessary. For example, it follows from (4.14) that f (x; u0 (x); ¢) is strictly convex at Q = Du0 (x) for x 2 « · , which is stronger than the necessary condition of quasiconvexity (cf. x 1). Note that even when f fails to satisfy (4.14), it might still be possible that this condition is veri ed by f~ = f + g, where g : « £ RN £ RN£n ! R is a null Lagrangian, that is, the integral Z g(x; u; Du) dx N £n
«
depends only on the boundary values of u. The key observation is that unlike the case when either of n or N = 1, in the multi-dimensional setting, a null Lagrangian is not necessarily an a¯ ne function of the gradient (cf. [2, 5]). In this case, theorem 3.3 can be translated to the multi-dimensional setting without the requirement
A. Taheri
182
of f being convex in the gradient argument (cf. theorem 5.2). But, of course, the di¯ culty would be to nd the appropriate g. We recall that in [29], Sivaloganathan shows that for a special class of non-convex functions f one can establish (4.14) for a modi ed f~ by nding a suitable null Lagrangian. He then employs this idea together with some machinery from Hamilton{Jacobi theory to establish a local stability result in nonlinear elasticity. We will later see how this follows from theorem 3.3.
5. Local stability theorems The positivity of the second variation of I at the stationary point u0 is a key assumption in the su¯ ciency theorems 3.1 and 3.3. In this section we show that under reasonable convexity assumptions on f this can always be `locally’ true. We then apply this observation to prove local stability of stationary points. In particular, we are able to obtain the result of Sivaloganathan [29] without any need for the construction of local elds and the Hamilton{Jacobi theory (cf. also Zhang [34, 35]). To start, let us recall that a C 2 function f : RN £n ! R is strongly elliptic at A 2 RN£n if and only if there exists ¬ > 0 such that D2 f (A)(¶ « · ; ¶ « · ) > ¬ j¶ j2 j· j2 for all ¶ 2 RN and · 2 Rn . Proposition 5.1. Let f 2 C 2 (« · £ RN £ RN £n ; R), u0 2 C 1 (« · ; RN ), x0 2 « and f (x0 ; u0 (x0 ); ¢) to be strongly elliptic at Du0 (x0 ). Then there exist ½ ; ¯ > 0 such that ¯ 2 I(u0 ; ’) > ½ k’k2W 1;2 (B¯
;RN )
for all ’ 2 W01;2 (B¯ ; RN ). Proof. Let us set fijkl (x) := fPij Pkl (x; u0 (x); Du0 (x)). It follows from the strong ellipticity condition on f and an application of Plancherel’s theorem (cf. [33]) that Z
Z jD’j2 dx
fijkl (x0 )’i;j ’k;l dx > ¬ «
«
for some ¬ = ¬ (x0 ) > 0 and all ’ 2 W 01;2 (« ; RN ). Moreover, a simple continuity argument shows that Z Z fijkl (x)’i;j ’k;l dx > fijkl (x0 )’i;j ’k;l dx « « Z j(fijkl (x) fijkl (x0 ))’i;j ’k;l j dx Z « > 12 ¬ jD’j2 dx; «
Su± ciency theorems for local minimizers
183
provided ’ vanishes outside a su¯ ciently small ball around x0 . Hence Z ¯ 2 I(u0 ; ’) = (fijkl (x)’i;j ’k;l + 2fPij uk (x; u0 ; Du0 )’i;j ’k «
³
Z > B¯ (x0 )
jD’j2
1 ¬ 2
Z
>
(jD’j2
1 4¬
+ fuk ul (x; u0 ; Du0 )’k ’l ) dx ´ 1 2 2 C("jD’j + j’j Cj’j2 ) dx " C1 j’j2 ) dx;
B¯ (x0 )
where ’ 2 W 1;2 (« ; RN ) is assumed to vanish outside B¯ (x0 ) for some small enough ¯ > 0. By taking ¯ smaller if necessary, we can make the right-hand side in the last inequality strictly positive for non-zero ’. The result is now a consequence of proposition 4.7. Combining the above proposition together with theorem 3.3 and recalling remark 4.20, we can now state the following (cf. [29, theorems 2.4 and 3.2]). Theorem 5.2. Let f 2 C 2 (« · £RN £RN £n ; R) and u0 2 C 1 (« · ; RN ) be a stationary · r (x0 ) £ RN £ point of I. For given x0 2 « , let there be r; ½ > 0 and g 2 C 2 (B N £n R ; R) such that the integral Z g(x; u0 + ’; Du0 + D’) dx Br (x0 )
is well de¯ned and constant for all ’ 2 W 01;2 (Br (x0 ); RN ) with k’kL1
(Br (x0 );RN )
0 such that, for any variation ’ 2 W 1;2 (« ; RN ) vanishing outside B¯ (x0 ), I(u0 + ’) provided k’kL1
(« ;RN )
I(u0 ) > ® k’k2W 1;2 («
;RN ) ;
